An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds

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1 An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds Massimiliano Berti 1, Livia Corsi 1, Michela Procesi 2 1 Dipartimento di Matematica, Università di Napoli Federico II, Napoli, I-80126, Italy 2 Dipartimento di Matematica, Università di Roma La Sapienza, Roma, I-00185, Italy m.berti@unina.it, livia.corsi@unina.it, mprocesi@mat.uniroma1.it Abstract We prove an abstract Implicit Function Theorem with parameters for smooth operators defined on sequence scales, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups. Keywords: Quasi-periodic solutions for PDEs; Nash-Moser theory; small divisor problems; Nonlinear Schrödinger and wave equations; analysis on compact Lie groups MSC classification: 37K55; 58C15; 35Q55; 35L05 Contents 1 Introduction 2 2 An implicit function theorem with parameters on sequence spaces Linear operators on H s and matrices Main abstract results Applications to PDEs Analysis on Lie groups Proof of Theorem 1.1 for NLW Proof of Theorem 1.1 for NLS An abstract Nash-Moser theorem 24 1

2 4.1 Proof of Theorem Initialisation of the Nash-Moser scheme Iterative step Proof of Theorem Initialisation Inductive step Separation properties Regularity A Proof of the multiscale Proposition Introduction In the last years several works have been devoted to the search of quasi-periodic solutions of Hamiltonian PDEs in higher space dimensions, like analytic nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on T n. A major difficulty concerns the verification of the so-called Melnikov non-resonance conditions. The first successful approach, due to Bourgain [9], [10], used a Newton iterative scheme which requires only the minimal (first-order) Melnikov conditions, which are verified inductively at each step of the iteration. In this paper we prove an abstract, differentiable Nash-Moser Implicit Function Theorem with parameters for smooth operators defined on Hilbert sequence scales. As applications, we prove the existence of quasi-periodic solutions with Sobolev regularity of the forced nonlinear wave equation and the nonlinear Schrödinger equation u tt u + mu = εf(ωt, x, u), x M, (1.1) iu t u + mu = εf(ωt, x, u), x M, (1.2) where M is any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, namely there exists a compact Lie group which acts on M transitively and differentiably. In (1.1)-(1.2) we denote by the Laplace-Beltrami operator, the mass m > 0, the parameter ε > 0 is small, and the frequency vector ω R d is non-resonant, see (1.4)-(1.6) below. Examples of compact connected Lie groups are the standard torus T n, the special orthogonal group SO(n), the special unitary group SU(n), and so on. Examples of (compact) manifolds homogeneous with respect to a compact Lie group are the spheres S n, the real and complex Grassmannians, and the moving frames, namely, the manifold of the k-ples of orthonormal vectors in R n with the natural action of the orthogonal group O(n) and many others; see for instance [11]. The study of (1.1)-(1.2) on a manifold M which is homogeneous with respect to a compact Lie group G is reduced to that of (1.1)-(1.2) on the Lie group G itself. Indeed M is diffeomorphic to M = G/N where N is a closed subgroup of G and the Laplace-Beltrami operator on M can be identified with the Laplace-Beltrami operator on G, acting on functions invariant under N (see [7]-Theorem 2.7 and [17, 18, 23]). 2

3 Concerning regularity we assume that the nonlinearity f C q (T d M R; R), resp. f(ϕ, x, u) C q (T d M C; C) in the real sense (namely as a function of Re(u), Im(u)), for some q large enough. We also require that f(ωt, x, u) = u H(ωt, x, u), H(ϕ, x, u) R, u C, (1.3) so that the NLS equation (1.2) is Hamiltonian. We assume that the frequency ω has a fixed direction, namely ω = λω, λ I := [1/2, 3/2], ω 1 := d for some fixed diophantine vector ω, i.e. ω satisfies p=1 ω p 1, (1.4) ω l 2γ 0 l d, l Z d \ 0}. (1.5) For the NLW equation (1.1) we assume also the quadratic diophantine condition γ 0 ω i ω j p ij p d(d+1), p Zd(d+1)/2 \ 0} (1.6) 1 i,j d which is satisfied for all ω 1 1 except a set of measure O(γ 1/2 0 ), see Lemma 6.1 in [5]. The search of quasi-periodic solutions of (1.1)-(1.2) reduces to finding solutions u(ϕ, x) of (ω ϕ ) 2 u u + mu = εf(ϕ, x, u), iω ϕ u u + mu = εf(ϕ, x, u, u), (1.7) in some Sobolev space H s of both the variables (ϕ, x), see Section 2. Theorem 1.1. Let M be any compact Lie group or manifold which is homogeneous with respect to a compact Lie group. Consider the NLW equation (1.1), and assume (1.4)-(1.6); for the NLS equation (1.2)-(1.3) assume only (1.4)-(1.5). Then there are s, q R such that, for any f, f C q and for all ε [0, ε 0 ) with ε 0 > 0 small enough, there is a map u ε C 1 ([1/2, 3/2], H s ), sup u ε (λ) s 0, as ε 0, λ [1/2,3/2] and a Cantor-like set C ε [1/2, 3/2], satisfying meas(c ε ) 1 as ε 0, such that, for any λ C ε, u ε (λ) is a solution of (1.7), with ω = λω. Moreover if f, f C then the solution u ε (λ) is of class C both in time and space. Actually Theorem 1.1 is deduced by the abstract Implicit Function Theorems 2.16, 2.18 (and Corollary 2.17) on scales of Hilbert sequence spaces. We postpone their precise formulations to Section 2.2, since some preparation is required. Theorem 1.1 is a first step in the direction of tackling the very hard problem of finding quasiperiodic solutions for NLW and NLS on any compact Riemannian manifold, if ever true. This is an open problem also for periodic solutions. In the particular case that the manifold M = T n is a n-dimensional torus, Theorem 1.1 is proved in [4] for NLS, and, in [5], for NLW. So far, the literature about quasi-periodic solutions is restricted to NLS and NLW on tori (which are compact commutative Lie groups). The first results were proved for the interval [0, π] by Kuksin 3

4 [19, 20], Wayne [27], Pöschel [22, 21], or for the 1-dimensional circle T by Craig-Wayne [13], Bourgain [8], and Chierchia You [12]. For higher dimensional tori T n, n 2, the first existence results have been obtained by Bourgain [9, 10] for NLS and NLW with Fourier multipliers via a multiscale analysis, recently applied by Wang [26] for completely resonant NLS. Using KAM techniques, Eliasson-Kuksin [14, 15] proved existence and stability of quasi-periodic solutions for NLS with Fourier multipliers, see also Procesi-Xu [25]. Then Geng-Xu-You [16] proved KAM results for the cubic NLS in dimension 2 and Procesi-Procesi [24] in any dimension n and polynomial nonlinearity. The reason why these results are confined to tori is that these proofs require specific properties of the eigenvalues and the eigenfunctions must be the exponentials or, at least, strongly localized close to exponentials. Recently, Berti-Bolle [4, 5] have extended the multiscale analysis to deal with NLS and NLW on T d with a multiplicative potential. In such a case the eigenfunctions may not be localized close to the exponentials. In the previous paper [7], Berti-Procesi proved existence of periodic solutions for NLW and NLS on any compact Lie group or manifold homogenous with respect to a compact Lie group. Main difficulties concern the eigenvalues of the Laplace-Beltrami operator, with their unbounded multiplicity, and the rule of multiplications of the eigenfunctions. A key property which is exploited is that, for a Lie group, the product of two eigenfunctions is a finite linear combinations of them (as for the exponentials or the spherical harmonics). From a dynamical point of view, it implies, roughly speaking, that only finitely many normal modes are strongly coupled. Theorem 1.1 extends the result in [7] to the harder quasi-periodic setting. As already said, it is deduced by the abstract Implicit Function Theorems These results rely on the Nash-Moser iterative Theorem 4.2 and a multiscale inductive scheme for deducing tame estimates for the inverse linearised operators at each step of the iteration, see Section 5. A main advantage of Theorem 2.16 is that the Cantor-like set of parameters C ε in (2.33) for which a solution exists is defined in terms of the solution u ε, and it is not inductively defined as in previous approaches. This formulation completely decouples the Nash-Moser iteration from the discussion about the measure of the parameters where all the required non-resonance conditions are verified. The possibility to impose the non-resonance conditions through the final solution was yet observed in [3] (in a Lyapunov-Schmidt context) and in [2] for a KAM theorem. In the present case the Cantor set C ε is rather involved. Nevertheless we are able to provide efficient measure estimates in the applications. This simplifies considerably the presentation because the measure estimates are not required at each step. In conclusion, in order to apply Theorems , one does not need to know the multiscale techniques nor the Nash-Moser approach: they can be used as a black box. We believe that Theorems can be applied to several other cases. The abstract hypotheses can be verified by informations of the harmonic analysis on the manifold. In the case of compact Lie groups and homogeneous manifolds, Theorem 1.1 follows by using only the harmonic analysis in [7] (see Section 3), which stems from the informations on the eigenvalues and eigenspaces of the Laplace- Beltrami operator provided by the highest weight theory, see [23]. We find it convenient to use a Nash-Moser scheme because the eigenvalues of the Laplacian are highly degenerate and the second order Melnikov non resonance conditions required for the KAM reducibility scheme might not be satisfied. Informal presentation of the ideas and techniques. Many nonlinear PDEs (such as (1.7)) can be seen as implicit function equations of the form F (ε, λ, u) = 0, 4

5 having for ε = 0 the trivial solution u(t, x) = 0, for each parameter λ I. Clearly, due to the small divisors, the standard Implicit Function Theorem fails, and one must rely on some Nash-Moser or KAM quadratic scheme. They are rapidly convergent iterative algorithms based on the Newton method and hence need some informations about the invertibility of the linearisation L(ε, λ, u) of F at any function u close to zero. Due to the Hamiltonian structure, the operator L(ε, λ, u) is self-adjoint and it is easy to obtain informations on its eigenvalues, implying the invertibility of L(ε, λ, u) with bounds of the L 2 -norm of L 1 (ε, λ, u) for most parameters λ. However these informations are not enough to prove the convergence of the algorithm: one needs estimates on the high Sobolev norm of the inverse which do not follow only from bounds on the eigenvalues. Usually this property is implied by a sufficiently fast polynomial off-diagonal decay of the matrices which represent the inverse operators. In the case of the interval [0, π], the eigenvalues of L(ε, λ, u) are often distinct, a property which enables to diagonalise L(ε, λ, u) via a smooth change of variables (reducibility) implying very strong estimates of the inverse operator in high Sobolev norm. This method automatically implies also the stability of the solution. Unfortunately, the eigenvalues of are not simple already on T (a fortiori neither on T n, n 2), so that generalising these reducibility methods is complicated and strongly depends on the equation. For NLS it is obtained in [14]. However, the convergence of the Nash-Moser scheme only requires tame estimates of the inverse in high Sobolev norm (for instance like the one in (4.4)) which may be obtained under weaker spectral hypotheses. In the case of NLS and NLW on T n these estimates have been obtained in [4, 5] via a multiscale analysis on Sobolev spaces (see [10] in an analytic setting). Informally, the multiscale method is a way to prove an off-diagonal decay for the inverse of a finite-dimensional invertible matrix with off-diagonal decay, by using informations on the invertibility (in high norm) of a great number of principal minors of order N much smaller than the dimension of the matrix. The polynomial off-diagonal decay of a matrix implies that it defines a tame operator between Sobolev spaces. In this paper we extend these techniques also to the case of compact Lie groups and manifolds which are homogeneous with respect to a compact Lie group (in the latter case we lift-up the equation to the Lie group). Two key points concern 1. the matrix representation of a multiplication operator u bu, 2. the properties of the eigenvalues of the Laplace-Beltrami operator. The multiplication rules for the eigenfunctions, together with the numeration of the eigenspaces provided the highest weight theory, implies that the multiplication operator by a Sobolev function b H s (M) is represented in the eigenfunction basis as a block matrix with off-diagonal decay, as stated precisely in Lemmas 3.1, 3.2 (proved in [7]). The block structure of this matrix takes into account the (large) multiplicity of the degenerate eigenvalues of on M (several blocks could correspond to the same eigenvalue). This in principle could be a problem because one can not hope to achieve any off-diagonal decay property for the matrices restricted to such blocks. However, as in [7], we do not need such a decay, being sufficient to control only the L 2 -operator norm on these blocks. Interestingly, properties of this type have been used by Bambusi, Delort, Grébert, Szeftel [1] for Birkhoff normal form results of Klein-Gordon equations on Zoll manifolds (a main difficulty in [1] is to verify the Birkhoff normal form non-resonance conditions). Concerning item 2, the eigenvalues of the Laplace-Beltrami operator on a Lie group are very similar to those on a torus, as stated in (3.6). This enables to prove separation properties of clusters of 5

6 singular/bad sites (i.e. Fourier indices with a corresponding small divisor) à la Bourgain [9], [10]. Thanks to the off-diagonal decay property proved in item 1 such resonant clusters interact only weakly. As in the case of T n (where the eigenvalues of are j 2, j Z n ) one does not gather into the same cluster all the indexes corresponding to the same eigenvalue. The reason is that such clusters would not satisfy the needed separation properties. We now give some more detail about the proof. In the usual PDE applications the function spaces decompose as a direct sum of eigenspaces of L(0, λ, 0), each of them being a direct product of the exponentials (for the time-direction) and the eigenspaces of + m (space direction). Hence we decompose u = k u k with k = (l, j) Z d Λ + (l Z d is the time-fourier component and j Λ + the space-fourier component). In particular L(0, λ, 0) is a diagonal operator which is proportional to the identity on each eigenspace. Moreover the dependence on l appears only through a scalar function D j (ω l), see (2.24a). In Theorems we revisit in a more abstract way the strategy of [4, 5], obtaining a unified and more general result for smooth operators F (ε, λ, u) acting on a Hilbert scale of sequences spaces. These results are based on three hypotheses which allow the possibility of passing from L 2 -norm estimates to high Sobolev norm bounds for the inverse linearized operators. We try to explain the meaning of this assumptions: i. the linearised operator can be written as the sum of a diagonal part D = D(λ) (which is the linearised operator at ε = 0, u = 0) plus a perturbation which has off-diagonal decay and is Töplitz in the time indices (see Hypothesis 1), ii. a uniform lower bound for the derivative of D j (y) on the set where D j (y) is small (see Hypothesis 2), iii. an assumption on the length of chains of singular sites (see Hypothesis 3). Under these hypotheses Theorem 2.16 implies the existence, for ε sufficiently small, of a function u = u ε (λ) which is a solution of the equation F (ε, λ, u) = 0 for all λ in the Cantor-like set C ε in (2.33), which is defined only in terms of the eigenvalues of submatrices of L(ε, λ, u ε (λ)). Roughly speaking the set C ε is defined as the intersection of two families of sets: 1. the sets G N of parameters λ for which the N-truncation of L(ε, λ, u ε ) is invertible in L 2 with good bounds of the L 2 -norm of the inverse (see (2.34)), 2. the sets G 0 N of parameters λ for which the principal minors of order N having a small eigenvalue are separated (see (2.35)). Technically the sets of type 2 are defined by exploiting the time-covariance property (2.29) and analysing the complexity of the real parameter θ (see (2.25)) for which the N-truncation of the timetraslated matrix L(ε, λ, θ, u ε ) have a small eigenvalue: since ω = λω is diophantine these two definitions are equivalent. Finally we underline two main differences with respect to the abstract Nash-Moser theorem in [6]. The first is that the tame estimates (4.4) required for the inverse linearized operators are much weaker than in [6]. Note, in particular, that the tame exponent in (4.4) grows like δs (this corresponds to an unbounded loss of derivatives as s increases). This improvement is necessary to deal with 6

7 quasi-periodic solutions. The second difference is that in [6] the measure issue was not yet completely decoupled from the Nash-Moser iteration, as, on the contrary, it is achieved in this paper thanks to the introduction of the set C ε in (2.33). The paper is essentially self-contained. The Appendix A contains the proof of the multiscale proposition 5.8 which follows verbatim as in [4]. We have added it for the convenience of the reader. Acknowledgements. We thank L. Biasco, P. Bolle, C. Procesi for many useful comments. 2 An implicit function theorem with parameters on sequence spaces We work on a scale of Hilbert sequence spaces defined as follows. We start from an index set K := I A = Z d Λ + A (2.1) where Λ + Λ is contained in a r-dimensional lattice (in general not orthogonal) Λ := j R r : j = r p=1 } j p w p, j p Z (2.2) generated by independent vectors w 1,... w r R r. The set A is finite, and in the applications will be either A = 1} (for NLW) or A = 1, 1} (for NLS). Given k K we denote k = (i, a) = (l, j, a) Z d Λ + A, k = i := max( l, j ), j := j = max j p N. (2.3) p If A = 1} we simply write k = (l, j). j = We require that Λ + has a product structure, namely that r j p w p, j = p=1 r j pw p Λ + j = p=1 r p=1 j p w p Λ + if p min(j p, j p) j p max(j p, j p). (2.4) Condition (2.4) will be used only in order to prove Lemma It could be probably weakened. In the applications it is satisfied. To each j Λ + we associate a multiplicity d j N. Then, for s 0, we define the (Sobolev) scale of Hilbert sequence spaces H s := H s (K) := u =u k } k K, u k C d j : u 2 s := } k k K w 2s u k 2 0 < (2.5) where 0 denotes the L 2 -norm in C d j and the weights w k := max(c, 1, w k ) satisfy c k w k C k, k K, for suitable constants 0 < c C. In the applications the weights w k are related to the eigenvalues of the Laplacian, see Examples 1, 2 below. Remark 2.1. The abstract Theorem 2.16 does not require any bound on the multiplicity d j. In the applications we use the polynomial bound (3.4) for Lemmas 3.1, 3.2 and for the measure estimates. 7

8 For any B K we define the subspaces HB s := u H s : u k = 0 for k / B}. (2.6) If B is a finite set the space HB s = H B does not depend on s and it is included in s 0 H s. Finally, for k = (i, a), k = (i, a ) K we denote dist(k, k ) := 1, i = i, a a, i i, otherwise, (2.7) where i is defined in (2.3). Remark 2.2. In principle i i may not be in Z d Λ + because Λ + is not a lattice. However, since Λ + Λ we can always compute i i by considering i i Z d Λ. In order to avoid this problem we will extend our vectors by setting them to zero on (Z d Λ A) \ K. All the constants that will appear in the sequel may depend on the index set K, the weights w k and on s. We will evidence only the dependence on s. 2.1 Linear operators on H s and matrices Let B, C K. A bounded linear operator L : HB s Hs C is represented, as usual, by a matrix in (M } M B k C := k )k C,k B, M k k Mat(d j d j, C). (2.8) It is useful to evidence a bigger block structure. We decompose B = B B, B := Proj Z d Λ + B, B := Proj A B and C = C C, defined in the same way. Now, for i = (l, j) C, i = (l, j ) B, we consider the matrix M i } i} := M i,a i,a } a C,a B, M i } i} Mat( C d j B d j, C), where B, C denote the cardinality of B, C A respectively. v := v k } k C C, for i = (l, j) C, we set v i} := v i,a } a C. In the same way, given a vector Remark 2.3. The difference with respect to [4], [5] is that the dimension of the matrix blocks M i } i} may not be uniformly bounded. They are scalars for the NLW equation in [5] and, in [4], for NLS, at most 1 2, 2 1 or 2 2 matrices, because d j = d j = 1 and 1 B, C 2. We endow Mat( C d j B d j, C) with the L 2 -operator norm, which we denote 0. Note that whenever a multiplication is possible one has the algebra property. Definition 2.4. (s-decay norm) For any M M B C we define its s-norm M 2 s := K 1 [M(i)] 2 i 2s (2.9) i Z d Λ 8

9 where i := max(1, i ), sup h M } h} 0, i C B, [M(i)] := h h =i,h C, h B 0, i / C B, (2.10) and K 1 > 4 i Z d Λ i 2s 0. We denote by (M s ) B C MB C the set of matrices with finite s norm s. If B, C are finite sets then (M s ) B C = MB C does not depend on s, and, for simplicity, we drop the apex s. Note that the norm s s for s s. The norm defined in (2.9) is a variation of that introduced in Definition 3.2 of [4]. The only difference concerns the dimensions of the blocks M i } i} as noted in Remark 2.3. However, since the matrices M i } i} are measured with the operator norm 0 the algebra and interpolation properties of the norm s follow similarly to [4], as well as all the properties in section 3-[4]. Indeed, given M M B C we introduce the Töpliz matrix which has the same decay norm M := (M i } i} ) MB C, M i } i} := [M(i i )]1 C dj B d j (2.11) M s = M s. (2.12) Lemma 2.5. Let M 1 M C D and M 2 M B C. Then M 1M 2 M B D satisfies M 1M 2 s M 1 M 2 s. Proof. For i B, i D, we have (M 1 M 2 ) i } i} (M 1 ) q} (2.10) (M2 0 i} ) i } 0 q} 1 )(i q)][(m 2 )(q i 0 q C q C[(M )] = q C[(M 1 )(i q)][(m 2 )(q i )] 1 D dj B d j 0 = q C[M 1 (i q)]1 D dj C d jq [M 2 (q i )]1 C djq B d j (2.11) = 0 (M 1 M 2 ) i } i}. 0 Therefore [(M 1 M 2 )(i i )] (M 1 M 2 ) i } i} 0 and the lemma follows. In what follows we fix s s 0 > (d + r)/2. Lemma 2.6. (Interpolation) For all s s 0 there is C(s) > 1 with C(s 0 ) = 1 such that, for any subset B, C, D K and for all M 1 M C D, M 2 M B C, one has M 1 M 2 s 1 2 M 1 s0 M 2 s + C(s) 2 M 1 s M 2 s0. (2.13) In particular, one has the algebra property M 1 M 2 s C(s) M 1 s M 2 s. Proof. For the Töpliz matrices M 1, M 2 the interpolation inequality (2.13) follows as usual (with C(s 0 ) 1 possibly taking K 1 larger). Hence Lemma 2.5 and (2.12) imply (2.13). The s norm of a matrix also controls the s norm (see [4]-Lemma 3.5). 9

10 Lemma 2.7. For any B, C K, let M M B C. Then Mh s C(s) M s0 h s + C(s) M s h s0, h H s B. (2.14) Proof. Regarding a vector h = h k } k B B as a column matrix, its s-decay norm is h 2 s = K 1 i B i 2s h i} 2 0. Hence (2.14) follows by Lemma 2.6 because c(s) h s h s c (s) h s. We conclude this section stating further properties of the s-norm: such lemmata are proved word by word as Lemmas 3.6, and 3.9 of [4] respectively. Lemma 2.8. (Smoothing) Let M M B C and N 2. For all s s 0 the following hold. (i) If M k k = 0 for all dist(k, k) < N (recall the definition (2.7)), then M s N (s s) M s. (2.15) (ii) If M k k = 0 for all dist(k, k) > N, then M s N s s M s, M s N s+d+r M 0. (2.16) Lemma 2.9. (Decay along lines) Let M M B C and denote by M k, k C, its k-th line. Then Lemma Let M M B C. Then M 0 M s0. M s C K 2 max k C M k s+d+r, s 0. (2.17) Definition We say that a matrix M M B C is left invertible if there exists N MC B NM = 1 B. In such a case N is called a left inverse of M. such that A matrix M is left-invertible if and only if it is injective. The left inverse is, in general, not unique. In what follows we shall denote by [ 1] M any left inverse of M when this does not causes ambiguity. Lemma (Perturbation of left-invertible matrices) Let M M B C be a left invertible matrix. Then for any P M B C such that [ 1] M s0 P s0 1/2 there exists a left inverse of M + P such that [ 1] (M + P ) [ 1] s0 2 M for any s s 0. Moreover, if [ 1] 2.2 Main abstract results We consider a non-linear operator [ 1] s0, (M + P ) s C(s) ( [ 1] M [ 1] s + M 2 ) s 0 P s, (2.18) M 0 P 0 1/2, then there is a left inverse of M +P which satisfies [ 1] (M + P ) 0 2 [ 1] M 0. (2.19) F (ε, λ, u) = D(λ)u + εf(u) (2.20) where ε > 0 is small, the parameter λ I [1/2, + ), and D(λ) is a diagonal linear operator D(λ) : H s+ν H s such that D(λ)h s, λ D(λ)h s C(s) h s+ν (2.21) 10

11 (in the applications D(λ) = iλω ϕ + m or (λω ϕ ) 2 + m) whose action on the subspace associated to a fixed index k is scalar, namely D(λ) = diag(d k (λ)1 dj ) k K. (2.22) We assume that, for some s 0 > (d + r)/2, the nonlinearity f C 2 (B s 0 1, Hs 0 ) (where B s 0 1 denotes the unit ball in H s 0 ) and the following tame properties hold: given S > s 0, for all s [s 0, S ) there exists a constant C(s) such that for any u s0 2, (f1) df(u)[h] s C(s) ( ) u s h s0 + h s, ) (f2) d 2 f(u)[h, v] s C(s) ( u s h s0 v s0 + h s v s0 + h s0 v s hold. Our goal is to find u = u ε (λ) H s for suitable s which solves the equation F (ε, λ, u ε (λ)) = 0 at least for some values of λ I. Then we assume further properties on the linearised operator L = L(ε, λ, u) = D(λ) + εt (u), D(λ) = diag(d k (λ)1 dj ) k K. (2.23) where T (u) is the matrix which represents the bounded linear operator df(u), see (2.8). Hypothesis 1. Let ω R d satisfy (1.5). There exists a function D : Λ + A R C and ν 0 > 0 such that (Covariance) D (l,j,a) (λ) = D j,a (λω l), λ I (2.24a) (Töplitz in time) T M K K : T (l,j,a ) (l,j,a) = T (j,a ) (j,a) (l l ) (2.24b) (Off-diagonal decay) T (u) s ν0 C(s)(1 + u s ), (2.24c) (Lipschitz) T (u) T (u ) s ν0 C(s)( u u s + ( u s + u s ) u u s0 ), (2.24d) for all u s0, u s0 2 and s 0 + ν 0 < s < S. For any θ R we set D(λ, θ) = Diag(D k (λ, θ)1 dj ), D k (λ, θ) := D j,a (λω l + θ) (2.25a) L(ε, λ, θ, u) := D(λ, θ) + εt (u). We need the following information about the unperturbed small divisors. (2.25b) Hypothesis 2. (Initialisation) There are n such that for all τ 1 > 1, N > 1, λ I, l Z d, j Λ +, a A, the set θ R : D(l,j,a) (λ, θ) N τ 1 } n I q intervals with meas(i q ) N τ 1. (2.26) We now distinguish which unperturbed small divisors are actually small or not. q=1 Definition (Regular/singular sites) We say that the index k K is regular for a matrix D := diag(d k 1 dj ), D k C, if D k 1, otherwise we say that k is singular. 11

12 We need an assumption which provides separation properties of clusters of singular sites. For any Σ K and j Λ + we denote the section of Σ at fixed j by Σ ( j) := k = (l, j, a) Σ}. Definition Let θ, λ be fixed and K > 1. We denote by Σ K any subset of singular sites of D(λ, θ) in K such that, for all j Λ +, the cardinality of the section Σ ( j) K satisfies #Σ( j) K K. Definition (Γ-Chain) Let Γ 2. A sequence k 0,..., k l K with k p k q for 0 p q l such that dist(k q+1, k q ) Γ, for all q = 0,..., l 1, (2.27) is called a Γ-chain of length l. Hypothesis 3. (Separation of singular sites) There exists a constant s and, for any N 0 2, a set Ĩ = Ĩ(N 0) such that, for all λ Ĩ, θ R, and for all K, Γ with KΓ N 0, any Γ-chain of singular sites in Σ K as in Definition 2.14, has length l (ΓK) s. In order to perform the multiscale analysis we need finite dimensional truncations of the matrices. Given a parameter family of matrices L(θ) with θ R and N > 1 for any k = (i, a) = (l, j, a) K we denote by L N,i (θ) (or equivalently L N,l,j (θ)) the sub-matrix of L(θ) centered at i, i.e. L N,i (θ) := L(θ) F F, F := k K : dist(k, k ) N}. (2.28) If l = 0, instead of the notation (2.28) we shall use the notation if also j = 0 we write and for θ = 0 we denote L N,j := L N,j (0). L N,j (θ) := L N,0,j (θ), L N (θ) := L N,0 (θ), By hypothesis 1, the matrix L = L(ε, λ, θ, u) has the following covariance property in time L N,l,j (ε, λ, θ, u) = L N,j (ε, λ, θ + λω l, u). (2.29) For τ 0 > 0, N 0 1 we define the set I := I(N 0, τ 0 ) := λ I : D k (λ) N τ 0 0 for all k = (i, a) K : i N 0 }. (2.30) Theorem Let e > d + r + 1. Assume that F in (2.20) satisfies (2.21)-(2.22), (f1) (f2) and Hypotheses 1, 2, 3 with S large enough, depending on e. Then, there are τ 1 > 1, N 0 N, s 1, S (s 0 + ν 0, S ν 0 ) with s 1 < S (all depending on e) and c(s) > 0 such that for all N 0 N 0, if the smallness condition εn S 0 < c(s) (2.31) holds, then there exists a function u ε C 1 (I, H s 1+ν ) with u 0 (λ) = 0, which solves F (ε, λ, u ε (λ)) = 0 (2.32) 12

13 for all λ C ε I defined in (2.33) below. The set C ε is defined in terms of the solution u ε (λ), as C ε := n 0 Ḡ 0 N 2n 0 ḠN 2n Ĩ I (2.33) 0 where Ĩ = Ĩ(N 0) is defined in Hypothesis 3, I in (2.30), and, for all N N, } Ḡ N := λ I : L 1 N (ε, λ, u ε(λ)) 0 N τ 1 /2, (2.34) ḠN 0 := λ I : j 0 Λ + there is a covering with B 0 N(j 0, ε, λ) N e q=1 B 0 N(j 0, ε, λ) := } (2.35) I q, with I q = I q (j 0 ) intervals with meas(i q ) N τ 1 } θ R : L 1 N,j 0 (ε, λ, θ, u ε (λ)) 0 > N τ 1 /2. (2.36) Finally, if the tame estimates (f1)-(f2), (2.24c), (2.24d) hold up to S = + then u ε (λ) s 0 H s. In applications, it is often useful to work in appropriate closed subspaces Ĥs (K) H s (K) which are invariant under the action of F. The following corollary holds: Corollary Assume, in addition to the hypotheses of Theorem 2.16, that F (ε, λ, ) : Ĥs+ν (K) Ĥ s (K), s > s 0. Then the function u ε provided by Theorem 2.16 belongs to C 1 (I, Ĥs 1+ν (K)). In Theorem 2.16 the Cantor like C ε defined in (2.33) may be empty. In order to prove that it has asymptotically full measure we need more informations. We fix N 0 = [ε 1/(S+1) ] so that the smallness condition (2.31) is satisfied for ε small enough. Theorem Let N 0 = [ε 1/(S+1) ] with ε small enough so that (2.31) holds. Assume, in addition to the hypotheses of Theorem 2.16, that for all N N 0, meas(i \ Ḡ0 N), meas(i \ ḠN) = O(N 1 ), Then C ε satisfies, for some K > 0, 1 meas(i \ (I Ĩ)) = O(N0 ). (2.37) meas(i \ C ε ) Kε 1/(S+1). (2.38) Proof. Let us denote N n = N0 2n. By the explicit expression (2.33) we have ( meas(i \ C ε ) = meas (ḠN n ) c Ĩc I c) which proves (2.38). n 0(Ḡ0 N n ) c n 0 meas(i \ Ḡ0 N n ) + meas(i \ ḠN n ) + meas(i \ (I Ĩ)) n 0 n 0 (2.37) C 0 n 0 N 1 n + C 1 N 1 0 C N 1 0 Kε1/(S+1) (2.39) In the applications to NLW and NLS the conditions (2.37) will be verified taking τ 0, τ 1 large, with a suitable e, see Proposition

14 3 Applications to PDEs Now we apply Theorems to the NLW and NLS equations (1.1)-(1.2). To be precise, when M is a manifold which is homogeneous with respect to a compact Lie group, we rely on Corollary We briefly recall the relevant properties of harmonic analysis on compact Lie groups that we need, referring to [23] (and [7]) for precise statements and proofs. A compact manifold M which is homogeneous with respect to a compact Lie group is, up to an isomorphism, diffeomorphic to M = G/N, G := G T r 2, (3.1) where G is a simply connected compact Lie group, T r 2 is a torus, and N is a closed subgroup of G. Then, a function on M can be seen as a function defined on G which is invariant under the action of N, and the space H s (M, C) (or H s (M, R)) can be identified with the subspace Ĥ s := Ĥs (G, C) := u H s (G) : u(x) = u(xg), x G = G T r 2, g N }. (3.2) Moreover, the Laplace-Beltrami operator on M can be identified with the Laplace-Beltrami operator on the Lie group G, acting on functions invariant under N (see Theorem 2.7, [7]). Then we lift the equations (1.1)-(1.2) on G and we use harmonic analysis on Lie groups. 3.1 Analysis on Lie groups Any simply connected compact Lie group G is the product of a finite number of simply connected Lie groups of simple type (which are classified and come in a finite number of families). Let G be of simple type, with dimension d and rank r. Denote by w 1,..., w r R r the fundamental weights of G and consider the cone of dominant weights Λ + (G) := j = r p=1 } j p w p : j p N Λ := j = r p=1 } j p w p : j p Z. Note that Λ + (G) satisfies (2.4) and indexes the finite dimensional irreducible representations of G. The eigenvalues and the eigenfunctions of the Laplace-Beltrami operator on G are µ j := j + ρ ρ 2 2, f j,σ (x), x G, j Λ + (G), σ = 1,..., d j, (3.3) where ρ := r i=1 w i, 2 denotes the euclidean norm on R r, and f j (x) is the (unitary) matrix associated to an irreducible unitary representation (R Vj, V j ) of G, precisely (f j (x)) h,k = R Vj (x)v h, v k, v h, v k V j, where (v h ) h=1,...,dimvj is an orthonormal basis of the finite dimensional euclidean space V j with scalar product,. We denote by N j the corresponding eigenspace of. The degeneracy of the eigenvalue µ j satisfies d j j + ρ d r 2. (3.4) The Peter-Weyl theorem implies the orthogonal decomposition L 2 (G) = N j. j Λ + (G) 14

15 Many informations on the eigenvalues µ j are known. There exists an integer Z N such that (see [7]-Lemma 2.6) the fundamental weights satisfy so that, in particular, w i w p Z 1 Z, i, p = 1,..., r, (3.5) µ j := j + ρ ρ 2 2, j 2 2, ρ j, ρ 2 2 Z 1 Z, j Λ + (G). (3.6) For a product group, L 2 (G 1 G 2 ) = L 2 (G 1 ) L 2 (G 2 ) and all the irreducible representations are obtained by the tensor product of the irreducible representations of G 1 and G 2. Hence we extend all the above properties to any compact Lie group G. For simplicity we still denote the dimension of the group as d and the rank as r. In particular Λ + (G) = Λ + (G) Z r 2 (see (3.1)) is the index set for the irreducible representations of G, with indices j = (j 1, j 2 ), j 1 Λ + (G), j 2 Z r 2, and ρ (ρ, 0). We denote the indices i = (l, j) Z d Λ + (G), so that L 2 (T d G T r 2 ) naturally decomposes as product of subspaces N k of the form We also set N k := e iϕ l N j = e iϕ l N j1 e ix 2 j 2. i := max( l, j ), l := l, j := j = max j i, i := max(1, i ). i The Sobolev spaces H s (T d G) and H s (T d G) H s (T d G), for a Lie group G, can be now identified with sequence spaces introduced in Section 2. Example 1. Let A := 1}, Λ + := Λ + (G) be the cone of fundamental weights and, for k = (l, j) K = Z d Λ +, let w k := l j + ρ 2 2. Then we may identify Hs (K) with the Sobolev space H s (T d G). Example 2. Let A := 1, 1}, Λ + := Λ + (G) be the cone of fundamental weights and, for k = (l, j, a) K := Z d Λ + A, let w k := l j + ρ 2 2. Then we may identify Hs (K) with the Sobolev space H s (T d G) H s (T d G). The final fundamental property that we exploit concerns the off-diagonal decay of the block matrix which represents the multiplication operator, see (2.8). The block structure of this matrix takes into account the (large) multiplicity of the degenerate eigenvalues of. We remark that several blocks could correspond to the same eigenvalue (as in the case of the torus). The next lemmas, proved in [7], are ultimately connected to the fact that the product of two eigenfunctions of the Laplace operator on a Lie group is a finite sum of eigenfunctions, see [7]-Theorem The forthcoming Lemmas are a reformulation of Proposition 2.19 and Lemma 7.1 in [7] respectively, and they require the polynomial bound (3.4). Lemma 3.1. ([7]-Proposition 2.19) Let K be as in Example 1 and b H s (T d G) be real valued. Then the multiplication operator B : u(ϕ, x) b(ϕ, x)u(ϕ, x) is self-adjont in L 2 and, for any s > (d+d)/2, B k k 0 where B k k Mat(d j d j, C), see (2.8). C(s) b s k k s (d+d)/2, k, k Z d Λ +, 15

16 Lemma 3.2. ([7]-Lemma 7.1) Let K be as in Example 2. Consider a, b, c H s (T d G) with a, b real valued. Then the multiplication operator with matrix ( ) a(ϕ, x) c(ϕ, x) B = c(ϕ, x) b(ϕ, x) is self-adjont in L 2 and, for any s > (d + d)/2, B i } i} 0 C(s) max( a s, b s, c s ) i i s (d+d)/2, i, i Z d Λ +. Corollary 3.3. Let B be a linear operator as in the previous two Lemmas. Then, for all s > (d+d)/2, B s C(s) max( a s+ν0, b s+ν0, c s+ν0 ), ν 0 := (2d + d + r + 1)/ Proof of Theorem 1.1 for NLW We apply Theorems to the operator F (ε, λ, ) : H s+2 (T d M, R) H s (T d M, R) u (λω ϕ ) 2 u u + mu εf(ϕ, x, u) which can be extended to H s+2 (T d G T r 2, R) H s (T d G T r 2, R) such that for all u H s+2 (T d ) Ĥs+2 one has F (ε, λ, u) H s (T d ) Ĥs where Ĥs is defined in (3.2). Setting A := 1}, Λ + := Λ + (G), G := G T r 2, we are in the functional setting of Example 1. The Hypothesis (2.21)-(2.22) holds with ν = 2 and the interpolation estimates (f1)-(f2) are verified provided that f(ϕ, x, u) C q for q large enough and s 0 > (d + d)/2 (d + r)/2. Remark 3.4. We require s 0 > (d + d)/2 in view of the embedding H s 0 (T d G) L (T d G) which, in turn, implies the algebra and interpolation properties of the spaces H s (T d G), s s 0. The weaker bound s 0 > (d + r)/2 is sufficient in order to prove the algebra and interpolation properties of the decay norm s (see Section 2.1), which hold with no constraint on the multiplicity d j. The linearised operator D(λ) εg(ϕ, x), D(λ) := (λ ω ϕ ) 2 + m, g(ϕ, x) := ( u f)(ϕ, x, u(ϕ, x)), is represented, in the Fourier basis e il ϕ f j (x), as in (2.23) with D k (λ) := D (l,j) (λ) = (λω l) 2 + m µ j (3.7) and T (u) is the matrix associated to the multiplication operator by g(ϕ, x). Corollary 3.3 implies that T (u) M s ν 0 for all u H s (T d G) and the estimates (2.24c), (2.24d) hold by interpolation. Hypothesis 1 holds with D j (y) = y 2 + m µ j. Also Hypothesis 2 holds: a direct computation shows that θ R : D (l,j) (λ, θ) N τ 1 } q=1,2 I q, I q intervals with meas(i q ) 4N τ 1 m µj + O(N 2τ 1 ), 16

17 and Hypothesis 2 holds with n = 16/ m. Hypothesis 3 about the length of chains of singular sites follows as in [5] because the eigenvalues of the Laplace-Beltrami operator are very similar to those on a torus, see (3.6). For γ > 0 let γ 1 + p Ĩ := λ Ĩ(γ) := [1/2, 3/2] : P (λω), non zero polynomial d(d+1) P (X) Z[X 1,..., X d ] of the form P (X) = p 0 + Lemma 3.5. For all N 0 2, Hypothesis 3 is satisfied with 1 i 1,i 2 d p i1,i 2 X i1 X i2 }. Ĩ defined in (3.8) and γ = N 1 0. Proof. The proof follows Lemma 4.2 of [5]. First of all, it is sufficient to bound the length of a Γ-chain of singular sites for D(λ, 0). Then we consider the quadratic form and the associated bilinear form Φ = Φ 1 + Φ 2 where (3.8) Q : R R r R, Q(x, j) := x 2 + j 2 2, (3.9) Φ 1 ((x, j), (x, j )) := xx, Φ 2 ((x, j), (x, j )) := j j. (3.10) For a Γ-chain of sites k q = (l q, j q )} q=0,...,l which are singular for D(λ, 0) (Definition 2.13) we have, recalling (3.7), (3.6), and setting x q := ω l q, Q(x q, j q + ρ) < 2 + m ρ 2 2, q = 0,..., l. Moreover, by (3.9), (2.27), we derive Q(x q x q0, j q j q0 ) C q q 0 2 Γ 2, 0 q, q 0 l, and so Φ((x q0, j q0 + ρ), (x q x q0, j q j q0 )) C q q 0 2 Γ 2. (3.11) Now we introduce the subspace of R 1+r given by S := Span R (x q x q0, j q j q0 ) : q = 0,..., l} and denote by s r + 1 the dimension of S. Let δ > 0 be a small parameter specified later on. We distinguish two cases. Case 1. For all q 0 = 0,..., l one has Span R (x q x q0, j q j q0 ) : q q 0 l δ, q = 0,..., l} = S. (3.12) In such a case, we select a basis f b := (x qb x q0, j qb j q0 ) = (ω l qb, j qb ), b = 1,..., s of S, where k qb = ( l qb, j qb ) satisfies k qb CΓ q b q 0 CΓl δ. Hence we have the bound f qb CΓl δ, b = 1,..., s. (3.13) Introduce also the matrix Ω = (Ω b b )s b,b =1 with Ωb b := Φ(f b, f b), that, according to (3.10), we write ( ) s Ω = Φ 1 (f b, f b ) + Φ 2 (f b, f b ) = X + Y, (3.14) b,b =1 17

18 where Xb b := (ω l q b )(ω l qb ) and Yb b := ( j q b ) ( j qb ). By (3.5) the matrix Y has entries in Z 1 Z and the matrix X has rank 1 since each column is ω l q1 X b = (ω l qb )., b = 1,..., s. ω l qs Then, since the determinant of a matrix with two collinear columns X b, X b, b b, is zero, we get P (ω) : = Z r+1 det(ω) = Z r+1 det( X + Y ) = Z r+1 (det(y ) det(x 1, Y 2,..., Y s )... det(y 1,..., Y s 1, X s )) which is a quadratic polinomial as in (3.8) with coefficients C(Γl δ ) 2(r+1). Note that P 0. Indeed, if P 0 then 0 = P (iω) = Z r+1 det(x + Y ) = Z r+1 det(f b f b ) b,b =1,...,s > 0 because f b } s b=1 is a basis of S. This contradiction proves that P 0. But then, by (3.8), the matrix Ω is invertible and Z r+1 det(ω) = P (ω) γ 1 + p d(d+1) γ (Γl δ ) C(d,r), (Ω 1 ) b b Cγ 1 (Γl δ ) C (d,r). (3.15) Now let S := S Φ := v R r+1 : Φ(v, f) = 0, f S}. Since Ω is invertible, the quadratic form Φ S is non-degenerate and so R r+1 = S S. We denote Π S : R r+1 S the projector onto S. Writing and since f b S, b = 1,..., s, we get Π S (x q0, j q0 + ρ) = w b := Φ ( (x q0, j q0 + ρ), f b ) = r+1 b =1 a b f b, (3.16) s a b Φ(f b, f b ) = b =1 s b =1 Ω b b a b where Ω is defined in (3.14). The definition of f b, the bound (3.11) and (3.12) imply w C(Γl δ ) 2. Hence, by (3.15), we deduce a = Ω 1 w C γ 1 (Γl δ ) C(r,d)+2, whence, by (3.16) and (3.13), Therefore, for any q 1, q 2 = 0,..., l, one has Π S (x q0, j q0 + ρ) γ 1 (Γl δ ) C (r,d). (x q1, j q1 ) (x q2, j q2 ) = Π S (x q1, j q1 + ρ) Π S (x q2, j q2 + ρ) γ 1 (Γl δ ) C 1(r,d), which in turn implies j q1 j q2 γ 1 (Γl δ ) C1(r,d) for all q 1, q 2 = 0,..., l. Since all the j q have r components (being elements of Λ + (G)) they are at most Cγ r (Γl δ ) C1(r,d)r. We are considering a Γ-chain in Σ K (see Definition 2.14) and so, for each q 0, the number of q 0,..., l} such that j q = j q0 is at most K and hence l γ r (Γl δ ) C 2(r,d) K (ΓK) r (Γl δ ) C 2(r,d) K l δc 2(r,d) (ΓK) r+c 2(r,d) 18

19 because of the condition ΓK N 0 (Hypothesis 3) and N 0 = γ 1. Choosing δ < 1/(2C 2 (r, d)) we get l (ΓK) 2(r+C 2(r,d)). Case 2. There is q 0 = 0,..., l such that dim(span R (x q x q0, j q j q0 ) : q q 0 l δ, q = 0,..., l}) s 1. Then we repeat the argument of Case 1 for the sub-chain (l q, j q ) : q q 0 l δ } and obtain a bound for l δ. Since this procedure should be applied at most r + 1 times, at the end we get a bound like l (ΓK) C 3(r,d). We have verified the hypotheses of Theorem 2.16 and Corollary The next proposition proves that also the assumptions (2.37) in Theorem 2.18 hold. Proposition 3.6. Fix τ 0 > r + 3d + 1, τ 1 > d + d + 2 and e = d + d + r + 4. There exists N 0 N (possibly larger than the N 0 found in Theorem 2.16) such that (2.37) holds. The proof of Proposition 3.6 -which will continue until the end of this section- follows by basic properties of the eigenvalues of a self-adjoint operator, which are a consequence of their variational characterisation. Proposition 3.6 is indeed a reformulation of Proposition 5.1 of [5]. With respect to [5], the eigenvalues of the Laplacian in (3.6) are different and the index set Λ is not an orthonormal lattice. Remark 3.7. There are two positive constants c < C such that c j j 2 C j. Hence if j > αc 1 N, N > 2 ρ 2, then the eigenvalues µ j in (3.6) satisfy µ j > α(α 1)N 2. On the other hand if j αc 1 N then µ j α(α + 1)(C/c) 2 N 2. Recall that if A, A are self-adjoint matrices, then their eigenvalues µ p (A), µ p (A ) (ranked in nondecreasing order) satisfy µ p (A) µ p (A ) A A 0. (3.17) We study finite dimensional restrictions of the the self-adjoint operator L(ε, λ) = L(ε, λ, u ε (λ)) = D(λ) + εt (ε, λ). One proceeds differently for j 0 (c + 5)c 1 N and j 0 < (c + 5)c 1 N. We assume N N 0 > 0 large enough and ε T 0 1. Lemma 3.8. For all j 0 Λ + (G), j 0 (c + 5)c 1 N, and for all λ [1/2, 3/2] one has B 0 N(j 0, ε, λ) N d+d+2 q=1 I q, with I q = I q (j 0 ) intervals with meas(i q ) N τ 1. Proof. We first show that B 0 N (j 0, ε, λ) R \ [ 2N, 2N]. Indeed by (3.17) all the eigenvalues µ l,j,σ (θ), σ = 1,..., d j, of L N,j0 (ε, λ, θ) (recall that d j denotes the degeneracy of the eigenvalues µ j in (3.3)), are of the form µ l,j,σ (θ) = δ l,j (θ) + O(ε T 0 ), δ l,j (θ) := (ω l + θ) 2 + m µ j. (3.18) Since ω 1 = λ ω 1 3/2, j j 0 N, l N, one has, by Remark 3.7, δ l,j (θ) ( 3 2 N + θ ) N 2 > 7N 2, θ < 2N. 19

20 By (3.18) we deduce µ l,j,σ (θ) 6N 2 and this implies BN 0 (j 0, ε, λ) [ 2N, 2N] =. Now set B 0,+ N := BN 0 (j 0, ε, λ) (2N, + ), B 0, N := B0 N (j 0, ε, λ) (, 2N). Since θ L N,j0 (ε, λ, θ) = diag l N, j j0 N 2(ω l + θ)1 dj N1, we apply Lemma 5.1 of [5] with α = N τ 1, β = N and E CN d+d (this is due to the bound d j j + ρ d r 2 ) and obtain B 0, N N d+d+1 q=1 We can reason in the same way for B 0,+ N I q, I q = I q (j 0 ) intervals with meas(i q ) N τ 1. and the lemma follows. Consider now j 0 < (c + 5)c 1 N. We obtain a complexity estimate for BN 0 (j 0, ε, λ) by knowing the measure of the set } B2,N(j 0 0, ε, λ) := θ R : L 1 N,j 0 (λ, ε, θ) 0 > N τ 1 /2. Lemma 3.9. For all j 0 < (c + 5)c 1 N and all λ [1/2, 3/2] one has B 0 2,N(j 0, ε, λ) I N := [ gn, gn], g := (2c + 8)Cc 1. Proof. If θ > gn one has ω l + θ θ ω l > (g (3/2))N > (2c + 6)Cc 1 N. Using Remark 3.7, all the eigenvalues proving the lemma. µ l,j,σ (θ) = (ω l + θ) 2 + m µ j + O(ε T 0 ) (Cc 1 N) 2, θ > gn, Lemma For all j 0 (c + 5)c 1 N and all λ [1/2, 3/2] one has B 0 N(j 0, ε, λ) ĈMN τ 1 +1 q=1 where M := meas(b 0 2,N (j 0, ε, λ)) and Ĉ = Ĉ(r). I q, I q = I q (j 0 ) intervals with meas(i q ) N τ 1 Proof. This is Lemma 5.5 of [5], where our exponent τ 1 is denoted by τ. Lemmas 3.9 and 3.10 imply that for all λ [1/2, 3/2] the set BN 0 (j 0, ε, λ) can be covered by N τ1+2 intervals of length N τ 1. This estimate is not enough. Now we prove that for most λ the number of such intervals does not depend on τ 1, showing that meas(b2,n 0 (j 0, ε, λ)) = O(N e τ 1 ) where e = d + d + r + 4 has been fixed in Proposition 3.6. To this purpose first we provide an estimate for the set } B 0 2,N(j 0, ε) := (λ, θ) [1/2, 3/2] R : L 1 N,j 0 (ε, λ, θ) 0 > N τ 1 /2. Then in Lemma 3.12 we use Fubini Theorem to obtain the desired bound for meas(b 0 2,N (j 0, ε, λ)). Lemma For all j 0 < (c+5)c 1 N one has meas(b 0 2,N (j 0, ε)) CN τ 1+d+d+1 for some C > 0. 20

21 Proof. Let us introduce the variables and set ζ = 1 λ 2, η = θ λ, (ζ, η) [4/9, 4] [ 2gN, 2gN] =: [4/9, 4] J N, (3.19) L(ζ, η) := λ 2 L N,j0 (ε, λ, θ) = diag l N, j j0 N Note that Then, except for (ζ, η) in a set of measure O(N τ 1+d+d+1 ) one has ( ( (ω l + η) 2 + ζ( µ j + m) ) 1 dj ) + εζt (ε, 1/ ζ). min µ j + m m. (3.20) j Λ + (G) L(ζ, η) 1 0 N τ 1 /8. (3.21) Indeed ( ) ε ζ L(ζ, η) = diag l N, j j0 N ( µj + m)1 dj + εt (ε, 1/ ζ) 2 ζ 1/2 λ T (3.20) m 2 1, for ε small (we used that ζ [4/9, 4]). Therefore Lemma 5.1 of [5] implies that for each η, the set of ζ such that at least one eigenvalue of L(ζ, η) has modulus 8N τ 1, is contained in the union of O(N d+d ) intervals with length O(N τ 1 ) and hence has measure O(N τ1+d+d ). Integrating in η J N we obtain (3.21) except in a set with measure O(N τ1+d+d+1 ). The same measure estimates hold in the original variables (λ, θ) in (3.19). Finally (3.21) implies L 1 N,j 0 (ε, λ, θ) 0 λ 2 N τ 1 /8 N τ 1 /2, for all (λ, θ) [1/2, 2/3] R except in a set with measure O(N τ 1+d+d+1 ). The same argument implies that where ḠN is defined in (2.34). Define the set F N (j 0 ) := meas([1/2, 3/2] \ ḠN) N τ 1+d+d+1 where Ĉ is the constant appearing in Lemma (3.22) λ [1/2, 3/2] : meas(b 0 2,N(j 0, ε, λ)) ĈN τ 1+d+d+r+2 } (3.23) Lemma For all j 0 (c + 5)c 1 N one has meas(f N (j 0 )) = O(N r 1 ). Proof. By Fubini Theorem we have meas(b 0 2,N(j 0, ε)) = Now, for any β > 0, using Lemma 3.11 we have CN τ 1+d+d+1 3/2 1/2 3/2 1/2 dλ meas(b 0 2,N(j 0, ε, λ)) dλ meas(b 0 2,N(j 0, ε, λ)). β meas(λ [1/2, 3/2] : meas(b 0 2,N(j 0, ε, λ)) β}) and for β = ĈN τ 1+r+d+d+2 we prove the lemma (recall (3.23)). 21

22 Lemma If τ 0 > r + 3d + 1 then meas([1/2, 3/2] \ I) = O(N0 1 ) where I is defined in (2.30). Proof. Let us write [1/2, 3/2] \ I = R l,j, R l,j := l, j N 0 λ Λ : (λω l) 2 + µ j m N τ 0 0 Since µ j + m m > 0, then R 0,j = if N 0 > m 1/τ 0. For l 0, using the Diophantine condition (1.5), we get meas(r l,j ) CN τ 0+2d 0, so that because τ 0 r 3d > 1. meas([1/2, 3/2] \ I) l, j N 0 meas(r l,j ) CN τ0+r+3d 0 = O(N 1 0 ) The measure of the set Ĩ in (3.8) is estimated in [5]-Lemma 6.3 (where Ĩ is denoted by G). Lemma If γ < min(1/4, γ 0 /4) (where γ 0 is that in (1.6)) then meas([1/2, 3/2] \ Ĩ) = O(γ). Proof of Proposition 3.6 completed. By the definition in (3.23) for all λ F N (j 0 ) one has meas(b2,n 0 (j 0, ε, λ)) < O(N τ1+r+d+d+2 ). Thus for any λ F N (j 0 ), applying Lemma 3.10 we have B 0 N(j 0, ε, λ) N r+d+d+4 q=1 I q, I q intervals with meas(i q ) N τ 1. But then, using also Lemma 3.8, we have that (recall (2.35) with e = r + d + d + 4) Hence, using Lemma 3.12, meas(i \ Ḡ0 N) [1/2, 3/2] \ Ḡ0 N j 0 (c+5)c 1 N j 0 (c+5)c 1 N F N (j 0 ). meas(f N (j 0 )) O(N 1 ) which is the first bound in (2.37). The second bound follows by (3.22) with τ 1 > d + d + 2. Finally, Lemmas 3.13 and 3.14 with γ = N 1 0 implies the third estimate in (2.37). }. 3.3 Proof of Theorem 1.1 for NLS In order to apply Theorems to the Hamiltonian NLS, we start by defining two extensions F(u, v), H(u, v) of class C q (T d M C 2 ; C) (in the real sense) of f(u) in such a way that F(u, u) = H(u, u) = f(u) and u F(u, u) = v H(u, u) R, u F(u, u) = u H(u, u) = v F(u, u) = v H(u, u) = 0 22